Waveform Design For The Integrated Sensing, Communication and Simultaneous Wireless Information And Power Transfer System

1. Introduction

The next-generation communication systems require the integration of additional functionalities to enhance system spectral efficiency, energy efficiency, and reduce system costs. Integrated sensing and communication systems (ISACS), which converge sensing and communication functionalities within a unified framework, have gained significant traction in recent years [1,2,3,4]. By co-locating sensing and communication tasks and sharing common resources such as antennas and signal processing algorithms, ISACS promise enhanced performance, reduced latency, and improved resource utilization compared to traditional separate implementations. However, the functionalities of ISACS systems still do not fully leverage the potential of wireless radio frequency signals, such as power transfer (PT).

Certain modern wireless applications impose energy transfer requirements on ISACS systems. One such example is battlefield unmanned aerial vehicle (UAV). In combat environments, UAVs operating under constrained resources need to simultaneously accomplish target perception, information transmission, and enhance endurance. One approach to addressing this challenge is the efficient and rational allocation of power in ISACS systems to improve energy efficiency. Some existing works have studied power allocation problems in ISACS systems. Another approach is equipping ISACS with energy harvesting devices to collect energy from received radio frequency (RF) signals, thereby extending the system’s operational lifespan. Regarding the latter, simultaneous wireless information and power transfer (SWIPT), first proposed in [5], has garnered significant attention in recent years [6,7,8]. The core of SWIPT is to use a single signal as a carrier for both information transmission and energy transfer. Communication nodes extract information from the received signal, while energy harvesting nodes collect energy from it. Clearly, inspired by ISACS, SWIPT still has the potential to expand perception capabilities and further improve system efficiency.

In ISACS, the transmitted waveform plays a crucial role in determining system performance. The waveforms based on orthogonal frequency-division multiplexing (OFDM) is a promising choice due to its simple processing framework and high processing gain. However, the performance of OFDM-based waveforms sharply deteriorates in high-speed mobile scenarios [9]. Next-generation communication systems also demand robustness in high-Doppler time-varying scenarios, as required for ISACS. Orthogonal Time Frequency Space (OTFS), a modulation scheme that modulates/demodulates information in the Doppler-delay (DD) domain, has been proposed in recent years [10,11,12]. OTFS represents high-Doppler time-varying channels sparsely in the DD domain. Previous studies have shown that OTFS offers significant communication performance improvement compared to OFDM in high-speed time-varying scenarios [11,12]. OTFS is also utilized in sensing systems and demonstrates excellent performance compared to OFDM [9,13,14]. Due to the aforementioned advantages, cyclic prefix OTFS (CP-OTFS) is adopted as the fundamental waveform in this work.

Currently, there is relatively limited research on the integration of ISACS with SWIPT. In [15], the Cramer-Rao lower bound (CRLB) of the estimation of the targets’ angle is employed as the optimization objective, the beamforming is designed with communication and power transfer quality of service (QoS) constraints in mind. In this work, we propose a framework for cyclic prefix (CP)-OTFS based integrated simultaneous wireless sensing, communication, and power transfer system (ISWSCPTS) that incorporates both search mode (SM) and joint work mode (JWM). This framework integrates the advantages of ISACS and SWIPT, enabling simultaneous perception of targets, information transmission, and energy transfer capabilities. The main contributions of this work are as follows:

• The framework of ISWSCPTS is proposed. This framework comprises two operational modes: SM and JWM. To enhance sensing capabilities for both SM and JWM, The CP-OTFS matched filter (MF) based target detection and parameters estimation (MF-TDaPE) algorithm is proposed. This algorithm exhibits superior performance to OFDM in high-speed scenarios.
• An CP-OTFS ambiguity function (AF) shaping (AFS) waveform design algorithm is proposed. Firstly, a novel DD domain AF for CP-OTFS is proposed. Subsequently, aiming to minimize the integrated sidelobe level (ISL) of the proposed AF while adhering to the QoS for communication and PT and constant modulus constraints, a non-convex CP-OTFS waveform design optimization problem is formulated. This problem is then solved through an iterative algorithm to obtain waveforms with superior AF characteristics and interference resilience.
• The semidefinite relaxation (SDR) beamforming design (SDR-BD) algorithm tailored for ISWSCPTS is proposed. Initially, a non-convex optimization problem is formulated with sensing QoS as the optimization objective and communication and PT QoS as constraints. Subsequently, the SDR technique is employed to solve this problem. The designed waveform optimizes perceptual capability while ensuring communication and PT performance.

The structure of the paper is outlined as follows: Section 2.1 introduces the principles of CP-OTFS. Section 2.2 elaborates on the framework principles, waveform design, and beamforming design. Section 3 presents simulation results and corresponding discussions. Finally, Section 4 concludes with a summary and outlines future work prospects.

2. Materials and Methods

Considering an CP-OTFS based ISWSCPTS equipped with a uniform linear array (ULA) with $N_t$ transmitting elements and a single receiving antenna to sense a single target, deliver information to a single communication node (CN), and transfer energy to a single energy receiving node (ERN). As shown in Figure 1. The ISWSCPTS transmits a single information-carrying CP-OTFS signal. Through this multifunctional signal, the ISWSCPTS receives the target echo to complete perception, the CN receives the signal to accomplish information collection, and the ERN receives the signal to complete energy harvesting. The ISWSCPTS is also referred to as sensing node (SN) due to its function of sensing. We details the system model of ISWSCPTS starting from the CP-OTFS model.

2.1. The CP-OTFS Model

2.1.1. Basic Concepts of CP-OTFS

The time–frequency (TF) plane is discretized to a $N\times M$ grid as following:

$$\Delta =(nT,m\Delta f),n=0,...,N-1,m=0,...,M-1$$

(1)

where T and $\Delta f=1/T$ are sampling intervals of time and frequency axes, respectively.

The modulated TF samples $\chi [n,m]$ are transmitted over an OTFS frame with duration $\widehat{T}=NT$ and occupy a bandwidth $B=M\Delta f$. The sampling frequency $f_s=B$.

The DD plane is discretized to an $N\times M$ lattice as following:

$$\Xi =(\frac{k}{NT},\frac{l}{M\Delta f}),k=0,...,N-1,l=0,...,M-1.$$

(2)

2.1.2. Input-Output Relationship of CP-OTFS Based ISWSCPTS

   A. The transmitter model

The CP-OTFS based system is depicted in Figure 2. The symbols $\chi [k,l]$ in DD domain is first transformed into symbols $\widehat{\chi }[n,m]$ in TF domain by the inverse symplectic finite Fourier transform (ISFFT). Then, the Heisenberg transform is applied to $\widehat{\chi }$ to generate time domain discrete signal $\widehat{x}\left[i\right],i=0,...,NM-1$. The expression of $\widehat{x}\left[i\right]$ is

$$\begin{array}{cc}\hfill \widehat{x}\left[i\right]=& \hfill \sum _{n^{\prime }}^{N-1}\sum _{m^{\prime }}^{M-1}\sum _{k^{\prime }}^{N-1}\sum _{l^{\prime }}^{M-1}\chi [k,l]\hfill \\ & \hfill \exp \left\{j2\pi (\frac{k^{\prime }{n}^{\prime }}{N}-\frac{l^{\prime }{m}^{\prime }}{M}+\frac{m^{\prime }i}{M})\right\}{g}_{tx}[i-n^{\prime }M],\hfill \end{array}$$

(3)

where the $g_{tx}\left[\iota \right]$ is the transmitting rectangular pulse and its expression is

$$g_{tx}\left[\iota \right]=\left\{\begin{array}{c}\hfill 1,0\le \iota \le M-1\hfill \\ \\ \hfill 0,otherwise\hfill \end{array}\right.$$

(4)

Next, the discrete CP-OTFS transmitting signal $x\left[i\right]$ is constructed by adding a CP of length M to $x\left[i\right]$. Finally, $x\left[i\right]$ is converted to radio frequency (RF) signal $x\left(t\right)$ by RF transmitter module.

   B. The channel model

In the application scenarios of ISWSCPTS, the perception target, communication channel, and energy transmission channel can all be modeled using a unified framework based on the DD channel (DDC) model. Assuming there are P DDCs which can be expressed as:

$$\rho (\tau ,f_d)=\sum _{p=1}^P{\alpha }_p\delta (\tau -{\tau }_p)\delta (f_d-f_{d_p}),$$

(5)

where ${\alpha }_p$ is the channel gain coefficient, $f_{d_p}$ is the Doppler, and ${\tau }_p$ is the delay. Let $\alpha ={[{\alpha }_1,...,{\alpha }_p]}^{\mathrm{T}}$.

In this work, we make the following assumptions:

$$\begin{array}{cc}\hfill & \hfill {\tau }_p=l_p/M\Delta f,l_p=0,1,...,M-1\hfill \\ \hfill & \hfill f_{d_p}={\left(k_p\right)}_N/NT,k_p=0,1,...,N-1\hfill \end{array}$$

(6)

where $l_p$ and $k_p$ are integers.

Thus, the discrete form of $\rho (\tau ,f_d)$ can be expressed as:

$$\rho [k,l]=\sum _{p=1}^P{\alpha }_p\delta [k-{\left(k_p\right)}_N]\delta [l-l_p]$$

(7)

where

$${\left(k_p\right)}_N=\left\{\begin{array}{c}\hfill k_p,k_p\le N/2\hfill \\ \\ \hfill N-k_p,otherwise\hfill \end{array}\right.$$

(8)

   C. The receiver model

The received signal $r\left(t\right)$ is transformed into the discrete-time signal $r\left[i\right],i=0,...,NM-1,$ by RF receiver module and removing the CP. $r\left[i\right]$ can be expressed as:

$$r\left[i\right]=\sum _{p=1}^P{\alpha }_{p}x\left[{[i-l_p]}_{NM}\right]\exp \left\{j2\pi f_{d_p}(\frac{iT}{M}+T)\right\}+\widehat{w}\left[i\right]$$

(9)

where ${[•]}_{NM}$ denotes modulo $NM$ operation, $\widehat{w}\left[i\right]$ is the additive Gaussian white noise (AWGN).

$r\left[i\right]$ is converted to received symbols $\widehat{{\rm Y}}[n,m]$ in TF domain through the Wigner transform. $\widehat{{\rm Y}}[n,m]$ can be expressed as:

$$\widehat{{\rm Y}}[n,m]=\sum _{i=0}^{NM-1}r\left[i\right]g_{rx}^{*}[i-nM]\exp \left(j2\pi \frac{mi}{M}\right)$$

(10)

where $g_{rx}\left[\iota \right]$ is the receiving rectangular pulse and its expression is

$$g_{rx}\left[\iota \right]=\left\{\begin{array}{c}\hfill 1,0\le \iota \le M-1\hfill \\ \\ \hfill 0,otherwise\hfill \end{array}\right.$$

(11)

Finally, $\widehat{{\rm Y}}[n,m]$ is converted into the received symbols ${\rm Y}[k,l]$ in DD domain through the symplectic finite Fourier transform (SFFT). ${\rm Y}[k,l]$ can be expressed as:

$${\rm Y}[k,l]=\sum _{n=0}^{N-1}\sum _{m=0}^{M-1}\widehat{{\rm Y}}[n,m]\exp \{-j2\pi (\frac{kn}{N}-\frac{lm}{M})\}+\mathbf{W}[k,l]$$

(12)

where $\mathbf{W}$ is AWGN matrix.

After laborious derivations, (12) can be recast as:

$${\rm Y}[k,l]=\sum _{k^{\prime }=0}^{N-1}\sum _{l^{\prime }=0}^{M-1}\chi [k^{\prime },l^{\prime }]H_{k,l}[k^{\prime },l^{\prime }]+\mathbf{W}[k,l]$$

(13)

where the expression of $H_{k,l}[k^{\prime },l^{\prime }]$ is as follows:

$$H_{k,l}[k^{\prime },l^{\prime }]=\frac{1}{N{M}^2}(H_{ISI}+H_{ICI})\exp \left(j2\pi f_{d}T\right)$$

(14)

In (14), $H_{ISI}$ and $H_{ICI}$ are expressed as follows:

(15)

(16)

According to (13), each receiving symbol is the result of summing all input symbols weighted by $H_{k,l}[k^{\prime },l^{\prime }]$. Clearly, $H_{k,l}[k^{\prime },l^{\prime }]$ is related to variables l, l, $k^{\prime }$, and $l^{\prime }$. The magnitude response of $H_{k,l}[k^{\prime },l^{\prime }]$ is as follows:

$$|H_{k,l}[k^{\prime },l^{\prime }]|=\left\{\begin{array}{c}\hfill 1,k^{\prime }={[k-k_p]}_N,l^{\prime }={[l-l_p]}_M\hfill \\ \hfill 0,otherwise\hfill \end{array}\right.$$

(17)

Therefore, the summation operation in (13) can be removed, and it can be simplified as follows:

$${\rm Y}[k,l]=\sum _{p=1}^P{\alpha }_p\chi [{[k-k_p]}_N,{[l-l_p]}_M]H_{k,l}[k_p,l_p]+\mathbf{W}[k,l],$$

(18)

where $H_{k,l}[k_p,l_p]$ is defined in (21), the definitions of $H_1$ and $H_2$ are (19) and (20), respectively. Note that (19)–(21) are exact expressions without any approximation.

$$\begin{array}{c}\hfill H_1=N\sum _{m=0}^{M-1}\sum _{m^{\prime }=0}^{M-1}\exp \{-j2\pi \frac{l(m^{\prime }-m)}{M}\}\sum _{i^{\prime }=l_p}^{M-1}\exp \left\{j2\pi \frac{m^{\prime }-m+f_{d_p}T}{M}\right\}\end{array}$$

(19)

$$\begin{array}{c}\hfill H_2=N\exp (-j2\pi \frac{{[k-k_p]}_N}{N})\sum _{m=0}^{M-1}\sum _{m^{\prime }=0}^{M-1}\exp \{-j2\pi \frac{l(m^{\prime }-m)}{M}\}\sum _{i^{\prime }=0}^{l_p-1}\exp \left\{j2\pi \frac{m^{\prime }-m+f_{d_p}T}{M}\right\}\end{array}$$

(20)

According to (3)–(12), after tedious derivations, ${\rm Y}[k,l]$ can be expressed as:

$$H_{k,l}[k_p,l_p]=\frac{1}{N{M}^2}(H_1+H_2)\exp \left(j2\pi f_{d_p}T\right)$$

(21)

2.2. Framework of CP-OTFS Based ISWSCPTS

ISWSCPTS encompasses two work modes: SM JWM for perception, communication, and PT. The following provides a detailed description of these two modes.

2.2.1. The SM of ISWSCPTS

In the SM, ISWSCPTS lacks a priori information of DDCs for the target, communication, and power transfer, necessitating estimation of these parameters. In this mode, the ISWSCPTS scans the region of interest in a phased-array manner to acquire states of DDCs from various angles.

The TF-domain received target echo whose angle is the same as the searching direction ${\mathbf{R}}_{SN}\in {\mathbb{C}}^{NM\times 1}$ can be expressed as:

$${\mathbf{R}}_{SN}=\sqrt{P_t}\alpha {\mathbf{a}}_t^{\mathrm{H}}\left({\theta }_{tar}\right)\eta {\mathbf{s}}_{SM}+{\mathbf{w}}_{SN},$$

(22)

where $P_t$ is the transmitting power, $\alpha$ is the target scattering coefficient (TSC), ${\mathbf{s}}_{SM}$ is the TF-domain transmitting signal in search mode, ${\mathbf{w}}_{SN}\in {\mathbb{C}}^{NM\times 1}$ is the AWGN matrix with the variance of each entry being ${\sigma }_{SN}^2$, ${\theta }_{tar}$ is the angle of the target, ${\mathbf{a}}_t\left({\theta }_{tar}\right)\in {\mathbb{C}}^{N_t\times 1}$ is the steering vector and ${\mathbf{a}}_t\left({\theta }_{tar}\right)$ can be expressed as:

$$\begin{array}{cc}\hfill {\mathbf{a}}_t\left({\theta }_{tar}\right)=& \hfill [1,\exp (-j2\pi \frac{dsin\left({\theta }_{tar}\right)}{{\lambda }_c}),\exp (-j2\pi 2\frac{dsin\left({\theta }_{tar}\right)}{{\lambda }_c}),...,\hfill \\ \hfill & \hfill \exp (-j2\pi (N_t-1)\frac{dsin\left({\theta }_{tar}\right)}{{\lambda }_c}){]}^T,\hfill \end{array}$$

(23)

where d is the antenna spacing, ${\lambda }_c$ represents the wavelength.

In the phased-array manner, $\eta ={\mathbf{a}}_t\left({\theta }_{tar}\right)$, thus (22) can be reformulated as:

$${\mathbf{R}}_{SN}=\sqrt{P_t}\alpha N_t{\mathbf{s}}_{SM}+{\mathbf{w}}_{SN},$$

(24)

According to (18), the vector form of DD-domain echo ${\mathbf{y}}_{SN}\in {\mathbb{C}}^{NM\times 1}$ can be expressed as:

$${\mathbf{y}}_{SN}=\sqrt{P_t}\alpha N_t({\mathbf{h}}_{SN}\odot {\widehat{\mathbf{x}}}_{SN})+{\mathbf{w}}_{SN},$$

(25)

where ${\mathbf{h}}_{SN}=\mathrm{vec}\left({\mathbf{H}}_{SN}\right)$, ${\widehat{\mathbf{x}}}_{SN}=\mathrm{vec}\left({\widehat{\mathbf{X}}}_{SN}\right)$, $\mathrm{vec}(•)$ vectorizes matrix along the row direction. For a target with delay tap $l_{tar}$ and Doppler tap $k_{tar}$, the $(k,l)$-th entry of ${\mathbf{H}}_{SN}$ can be expressed as:

$${\mathbf{H}}_{SN}[k,l]=H_{k,l}[k_{tar},l_{tar}].$$

(26)

the $(k,l)$-th entry of ${\mathbf{X}}_{SN}$ can be expressed as:

$${\widehat{\mathbf{X}}}_{SN}[k,l]=\chi [{[k-k_{tar}]}_N,{[l-l_{tar}]}_M].$$

(27)

The MF algorithm proposed in Reference [9] represents a state-of-the-art approach for OTFS-based target detection and parameter estimation. In this work, we offer a re-engineered version of the MF algorithm, employing an equivalent yet distinct methodology. Note that (25) can be reformulated as:

$${\mathbf{y}}_{SN}=\sqrt{P_t}\alpha N_t{\Lambda }_{SN}(k_{tar},l_{tar}){\mathbf{P}}_{SN}(k_{tar},l_{tar}){\mathbf{x}}_{SM}+{\mathbf{w}}_{SN},$$

(28)

where ${\mathbf{x}}_{SM}=\mathrm{vec}\left(\chi \right)$, ${\Lambda }_{SN}(k_{tar},l_{tar})=\mathrm{diag}\left({\mathbf{h}}_{SN}\right)$, ${\mathbf{P}}_{SN}(k_{tar},l_{tar})$ is the permutation matrix. The $(i,j)$-th entry of ${\mathbf{P}}_{SN}(k_{tar},l_{tar})$ can be expressed as:

$${\mathbf{P}}_{SN}(k_{tar},l_{tar})=\left\{\begin{array}{c}\hfill 1,j={[k-kp]}_{N}M+{[l-lp]}_M\hfill \\ \hfill 0,otherwise\hfill \end{array}\right.$$

(29)

where $i=kM+l$. Let ${\mathbf{Q}}_{SN}(k_{tar},l_{tar})={\Lambda }_{SN}(k_{tar},l_{tar}){\mathbf{P}}_{SN}(k_{tar},l_{tar})$, (28) can be expressed as:

$${\mathbf{y}}_{SN}=\sqrt{P_t}\alpha N_t{\mathbf{Q}}_{SN}(k_{tar},l_{tar}){\mathbf{x}}_{SM}+{\mathbf{w}}_{SN},$$

(30)

We construct a matrix $\Psi \in {\mathbb{C}}^{NM\times NM}$. For the i-th row of $\Psi$ where $i=kM+l$, $\Psi (i,:)={\left(\frac{1}{\sqrt{P_t}{N}_t{P}_{SM}}{\mathbf{Q}}_{SN}(k,l){\mathbf{x}}_{SM}\right)}^H$, $P_{SM}={\mathbf{x}}_{SM}^H{\mathbf{x}}_{SM}$ is the power of transmitting signal. Then, the output ${\xi }_{SN}\in {\mathbb{C}}^{NM\times 1}$ of proposed MF algorithm can be expressed as:

$${\xi }_{SN}=\Psi {\mathbf{y}}_{SN}.$$

(31)

Peaks will appear in the indices corresponding to $k_{tar}$ and $l_{tar}$ in $|{\xi }_{SN}|$, hence target presence can be detected through the constant false alarm rate (CFAR) algorithm. Additionally, by utilizing $i=kM+l$, the delay and Doppler frequency of the target can be estimated. Specifically, if the $\widehat{i}$-th element of $|{\xi }_{SN}|$ is a peak, then the corresponding estimated delay tap ${\widehat{l}}_{tar}$, Doppler tap ${\widehat{k}}_{tar}$ and TSC $\widehat{\alpha }$ are:

$${\widehat{l}}_{tar}={\left[\widehat{i}\right]}_M,$$

(32)

$${\widehat{l}}_{tar}=⌊\frac{\widehat{i}}{M}⌋,$$

(33)

$$\widehat{\alpha }={\xi }_{SN}\left[\widehat{i}\right],$$

(34)

where $⌊•⌋$ is the floor operation. Note that $\Psi$ can be precomputed offline. The CP-OTFS MF based target detection and parameters estimation (MF-DaPE) algorithm is summarized in Algorithm 1.

Algorithm 1: The CP-OTFS MF-DaPE algorithm

Input: $\Psi$, ${\mathbf{y}}_{SN}$ Output: ${\widehat{l}}_{tar}$, ${\widehat{k}}_{tar}$, $\widehat{\alpha }$ 1 Calculate ${\xi }_{SN}$ through (31) 2 Find peak index in $|{\xi }_{SN}|$ through CFAR algorithm 3 Calculate ${\widehat{l}}_{tar}$ through (32) 4 Calculate ${\widehat{k}}_{tar}$ through (33) 5 Calculate $\widehat{\alpha }$ through (34) 6 Return ${\widehat{l}}_{tar}$, ${\widehat{k}}_{tar}$ and $\widehat{\alpha }$.

2.2.2. The JWM of ISWSCPTS

In the JWM, ISWSCPTS accomplishes target tracking, communication, and power transfer. In this work mode, assuming an approximate target location and using the channel state information (CSI) of the CN and ERN as priors is reasonable, as target detection has already been achieved in the SM, and CSI can be obtained by transmitting pilot signals. The ISWSCPTS aims to achieve better performance in the JWM. In this work, we enhance perception performance while ensuring communication and power transfer performance by designing the transmitting CP-OTFS signal and beamforming.

   A. The receiving model of JWM

The DD-domain received echo of the target in TM can be expressed as:

$${\mathbf{z}}_{SN}=\sqrt{P_t}\alpha {\mathbf{a}}_t^{\mathrm{T}}\left({\theta }_{tar}\right)\eta {\mathbf{Q}}_{SN}(k_{tar},l_{tar}){\mathbf{x}}_{JWM}+{\mathbf{w}}_{SN},$$

(35)

Similarly, the DD-domain received signal in CN can be expressed as follows:

$${\mathbf{z}}_{CN}=\sqrt{P_t}\beta {\mathbf{a}}_t^{\mathrm{T}}\left({\theta }_{CN}\right)\eta \mathbf{Q}(k_{CN},l_{CN}){\mathbf{x}}_{JWM}+{\mathbf{w}}_{CN},$$

(36)

where $\beta$ is the complex channel gain, ${\mathbf{a}}_t\left({\theta }_{IR}\right)\in {\mathbb{C}}^{N_t\times 1}$ is the steering vector corresponding to angle of CN, ${\mathbf{Q}}_{CN}$ is the communication channel response matrix with delay tap $l_{CN}$ and Doppler tap $k_{CN}$, ${\mathbf{w}}_{CN}\in {\mathbb{C}}^{NM\times 1}$ is the AWGN matrix with the variance of each entry being ${\sigma }_{CN}^2$.

The DD-domain received signal in ERN can be expressed as follows:

$${\mathbf{z}}_{ERN}=\sqrt{P_t}\gamma {\mathbf{a}}_t^{\mathrm{T}}\left({\theta }_{ERN}\right)\eta \mathbf{Q}(k_{ERN},l_{ERN}){\mathbf{x}}_{JWM}+{\mathbf{w}}_{ERN},$$

(37)

where $\gamma$ is the power transfer channel gain, ${\mathbf{a}}_t\left({\theta }_{ERN}\right)\in {\mathbb{C}}^{N_t\times 1}$ is the transmitting steering vector, ${\mathbf{Q}}_{ERN}$ is the communication channel response matrix with delay tap $l_{ERN}$ and Doppler tap $k_{ERN}$, ${\mathbf{w}}_{ERN}\in {\mathbb{C}}^{NM\times 1}$ is the AWGN matrix with the variance of each entry being ${\sigma }_{ERN}^2$.

   B. Waveform design by ambiguity function shaping

The ambiguity function (AF) is crucial metrics for assessing radar signals. To enhance the target tracking performance of the system, we formulate an optimization problem aimed at reshaping the AF to achieve lower integral side-lobe levels (ISL). The definition of the traditional radar signal ambiguity function is as follows:

$$A(\tau ,f_d)=\int x\left(t\right)x^{*}(t-\tau )\exp \left(j2\pi f_{d}t\right)dt,$$

(38)

where $x\left(t\right)$ is the transmitting signal, $\tau$ is the delay and $f_d$ is the Doppler frequency. However, the output signal of the OTFS system belongs to the delay-Doppler (DD) domain, and the traditional time-domain approach cannot be used to define the OTFS ambiguity function. For OTFS, a discrete AF in the DD domain has been proposed, with the expression as follows:

$${\mathcal{A}}_{k_0,l_0}[k,l]={\left|{\mathbf{x}}^{\mathrm{H}}\mathbf{Q}(k,l)\mathbf{Q}(k_0,l_0)\mathbf{x}\right|}^2,$$

(39)

where $\mathbf{x}$ is the DD-domain transmitting signal, $l_0$ and $k_0$ are delay tap and Doppler tap of interest, respectively. The ISL of ${\mathcal{A}}_{k_0,l_0}[k,l]$ is:

$$\mathrm{ISL}=\sum _{k=0,k\ne k_0}^{N-1}\sum _{l=0,l\ne l_0}^{M-1}{\left|{\mathbf{x}}^{\mathrm{H}}\mathbf{Q}(k,l)\mathbf{Q}(k_0,l_0)\mathbf{x}\right|}^2,$$

(40)

Thus, the problem of waveform design for AF shaping is expressed as:

$$P_0:\begin{array}{cc}\hfill \underset{{\mathbf{x}}_{JWM}}{min}& \hfill \sum _{k=0,k\ne k_0}^{N-1}\sum _{l=0,l\ne l_0}^{M-1}{\left|{\mathbf{x}}_{JWM}^{\mathrm{H}}\mathbf{Q}(k,l)\mathbf{Q}(k_0,l_0){\mathbf{x}}_{JWM}\right|}^2\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill |x_{JWM}\left(i\right)|=1,i=1,2,...,NM\hfill \end{array}$$

(41)

The constraint of (41) is constant modulus constraint, which is preferred by radar system. Based on the proposition 1 presented in [16], $P_0$ can be handled by sequentially solving the following approximation problem:

$$P_{t+1}:\begin{array}{cc}\hfill \underset{{\mathbf{x}}_{JWM}}{max}& \hfill u({\mathbf{x}}_{JWM},{\mathbf{x}}_{JWM}^{\left(t\right)})\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill |x_{JWM}\left(i\right)|=1,i=1,2,...,NM\hfill \end{array}$$

(42)

where ${\mathbf{x}}_{JWM}^{\left(t\right)}$ denotes the t-th iteration solution of the proposed iteration algorithm. The expression of $u({\mathbf{x}}_{JWM},{\mathbf{x}}_{JWM}^{\left(t\right)})$ is:

$$u({\mathbf{x}}_{JWM},{\mathbf{x}}_{JWM}^{\left(t\right)})={\mathbf{x}}_{JWM}^{\mathrm{H}}({\lambda }^{\left(t\right)}{\mathbf{I}}_{NM}-\Theta \left({\mathbf{x}}_{JWM}^{\left(t\right)}\right)){\mathbf{x}}_{JWM},$$

(43)

where ${\lambda }^{\left(t\right)}$ satisfies that ${\lambda }^{\left(t\right)}\ge \beta {\lambda }_{\max }\left(\Theta \left({\mathbf{x}}_{JWM}^{\left(t\right)}\right)\right)$. Let ${\Gamma }_{k_0,l_0}(k,l)=\mathbf{Q}(k,l)\mathbf{Q}(k_0,l_0)$, $\Theta \left(\mathbf{x}\right)$ is expressed as:

$$\Theta \left(\mathbf{x}\right)=\frac{1}{2(NM-1)}\sum _{k=0,k\ne k_0}^{N-1}\sum _{l=0,l\ne l_0}^{M-1}({\Gamma }_{k_0,l_0}{(k,l)}^{\mathrm{H}}\mathbf{x}{\mathbf{x}}^{\mathrm{H}}{\Gamma }_{k_0,l_0}(k,l)+{\Gamma }_{k_0,l_0}(k,l)\mathbf{x}{\mathbf{x}}^{\mathrm{H}}{\Gamma }_{k_0,l_0}{(k,l)}^{\mathrm{H}}).$$

(44)

Furthermore, according to proposition 2 in [16], $P_{t+1}$ can be solved through the following problem:

$$P_1:\begin{array}{cc}\hfill \underset{{\widehat{\mathbf{x}}}_{JWM}}{max}& \hfill \mathrm{Re}\left({\widehat{\mathbf{x}}}_{JWM}^{\mathrm{H}}{\mathbf{v}}^{\left(t\right)}\right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill |{\widehat{x}}_{JWM}\left(i\right)|=1,i=1,2,...,NM\hfill \end{array}$$

(45)

where ${\mathbf{v}}^{\left(t\right)}$ can be expressed as:

$${\mathbf{v}}^{\left(t\right)}=({\lambda }^{\left(t\right)}{\mathbf{I}}_{NM}-\Theta \left({\mathbf{x}}_{JWM}^{\left(t\right)}\right)){\mathbf{x}}_{JWM}^{\left(t\right)}.$$

(46)

Then, the closed-form solution for $P_1$ is:

$${\widehat{\mathbf{x}}}_{JWM}=\exp \left(j\arg \left({\mathbf{v}}^{\left(t\right)}\right)\right),$$

(47)

where $\exp (•)$ and $\arg (•)$ are applied element-wise to the vectors.

Note that the optimal waveform ${\mathbf{x}}_{JWM}^{★}$ can also be utilized in SM. This waveform design algorithm is called CP-OTFS AFS, and it is summarized in Algorithm 2.

Algorithm 2: CP-OTFS AFS

Input: Initial ${\mathbf{x}}_{JWM}^{\left(0\right)}$ and stop condition $ϵ$ Output: The optimized waveform ${\mathbf{x}}_{JWM}^{★}$ 1 Let $t\leftarrow 0$ and ${\mathbf{x}}_{JWM}^{\left(t\right)}\leftarrow {\mathbf{x}}_{JWM}^{\left(0\right)}$ 2 ${\mathbf{v}}^{\left(t\right)}\leftarrow ({\lambda }^{\left(t\right)}{\mathbf{I}}_{NM}-\Theta \left({\mathbf{x}}_{JWM}^{\left(t\right)}\right)){\mathbf{x}}_{JWM}^{\left(t\right)}$ 3 ${\widehat{\mathbf{x}}}_{JWM}\leftarrow \exp \left(j\arg \left({\mathbf{v}}^{\left(t\right)}\right)\right)$ 4 $t\leftarrow t+1$ 5 ${\mathbf{x}}_{JWM}^{\left(t\right)}\leftarrow {\widehat{\mathbf{x}}}_{JWM}$ 6 If $|{\mathbf{x}}_{JWM}^{\left(t\right)}-{\mathbf{x}}_{JWM}^{(t-1)}|\le ϵ$, return ${\mathbf{x}}_{JWM}^{★}={\mathbf{x}}_{JWM}^{\left(t\right)}$. Otherwise, return to step 2.

C. Beamforming Design for ISWSCPTS

In this section, we investigate the beamforming design for ISWSCPTS. Within the beamforming design challenges of ISWSCPTS, two key considerations emerge: 1) optimization of sensing performance; 2) fulfilling the requirements for power transfer in the context of ERN considerations.

We begin by deriving metrics for sensing, communication, and power transfer. Subsequently, we formulate the optimization problem for ISWSCPTS beamforming design. Finally, we propose an algorithm to solve the beamforming design optimization problem.

C1. Sensing metric

In the JWM, the ISWSCPTS system necessitates continuous estimation of target parameters, and the precision of target parameter estimation is closely tied to the signal-to-noise ratio (SNR). Consequently, we employ SNR as the sensing metric.

According to (25), the ${\mathrm{SNR}}_{\mathrm{SN}}$ is

$$\begin{array}{cc}\hfill {\mathrm{SNR}}_{\mathrm{SN}}& \hfill =\frac{\mathbb{E}\left\{\right|\left|\sqrt{P_t}\alpha {\mathbf{a}}_t^{\mathrm{H}}\left({\theta }_{tar}\right)\eta ({\mathbf{h}}_{SN}\odot {\mathbf{x}}_{SN})\right|{|}_F^2\}}{\mathbb{E}\left\{\right|\left|{\mathbf{w}}_{SN}\right|{|}_F^2\}}.\hfill \end{array}$$

(48)

Based on the properties of $|{\mathbf{h}}_{SN}{\left(j\right)|}^2=1$ and $|{\mathbf{x}}_{SN}{\left(j\right)|}^2=1$, through a series of calculations, (48) can be simplified to:

$$\begin{array}{cc}\hfill {\mathrm{SNR}}_{\mathrm{SN}}& \hfill =\frac{P_t{\left|\alpha \right|}^2{\eta }^{\mathrm{H}}{\mathbf{a}}_r\left({\theta }_{tar}\right){\mathbf{a}}_r^H\left({\theta }_{tar}\right)\eta }{{\sigma }_{SN}^2}\hfill \\ \hfill & \hfill ={\eta }^{\mathrm{T}}{\mathbf{G}}_{\mathrm{SN}}\eta .\hfill \end{array}$$

(49)

where ${\mathbf{G}}_{\mathrm{SN}}=\frac{P_t{\left|\alpha \right|}^2}{{\sigma }_{SN}^2}{\mathbf{a}}_r\left({\theta }_{tar}\right){\mathbf{a}}_r^{\mathrm{H}}\left({\theta }_{tar}\right)$.

C2. Communication metric

Due to the impact of communication quality, such as bit error rate (BER) and channel capacity, being closely related to SNR, and in order to maintain consistency with snesing metrics, SNR is also employed as the communication metric. Similar to derivation of sensing metric, the ${\mathrm{SNR}}_{\mathrm{CN}}$ is

$$\begin{array}{cc}\hfill {\mathrm{SNR}}_{\mathrm{CN}}& \hfill =\frac{\mathbb{E}\left\{\right|\left|\sqrt{P_t}\beta {\mathbf{a}}_t^{\mathrm{H}}\left({\theta }_{CN}\right)\eta {({\mathbf{h}}_{CN}\odot {\mathbf{x}}_{CN})}^{\mathrm{T}}\right|{|}_F^2\}}{\mathbb{E}\left\{\right|\left|{\mathbf{w}}_{CN}\right|{|}_F^2\}}\hfill \\ \hfill & \hfill =\frac{P_t{\left|\beta \right|}^2{\eta }^{\mathrm{H}}{\mathbf{a}}_t\left({\theta }_{CN}\right){\mathbf{a}}_t^{\mathrm{H}}\left({\theta }_{CN}\right)\eta }{{\sigma }_{CN}^2}\hfill \\ \hfill & \hfill ={\eta }^{\mathrm{H}}{\mathbf{G}}_{\mathrm{CN}}\eta ,\hfill \end{array}$$

(50)

where ${\mathbf{G}}_{\mathrm{CN}}=\frac{P_t{\left|\beta \right|}^2}{{\sigma }_{CN}^2}{\mathbf{a}}_t\left({\theta }_{CN}\right){\mathbf{a}}_t^{\mathrm{H}}\left({\theta }_{CN}\right)$.

C3. Power transfer metric

The ERN collects energy from signals emitted by the SN, thus the harvested energy is employed as the metric of Power transfer. Due to the negligible power of noise compared with transmitting signal, the harvested energy can be expressed as:

$$\begin{array}{cc}\hfill {\mathrm{E}}_{\mathrm{ERN}}& \hfill =\mathbb{E}\left\{\mu \right|\left|\sqrt{P_t}\gamma {\mathbf{a}}_t^{\mathrm{H}}\left({\theta }_{ERN}\right)\eta {({\mathbf{h}}_{ERN}\odot {\mathbf{x}}_{ERN})}^{\mathrm{T}}\right|{|}_F^2\}\hfill \\ \hfill & \hfill =\mu P_t{\left|\gamma \right|}^2{\eta }^{\mathrm{H}}{\mathbf{a}}_r\left({\theta }_{ERN}\right){\mathbf{a}}_r^{\mathrm{H}}\left({\theta }_{ERN}\right)\eta \hfill \\ \hfill & \hfill ={\eta }^{\mathrm{H}}{\mathbf{G}}_{\mathrm{ERN}}\eta ,\hfill \end{array}$$

(51)

where $\mu \in [0,1]$ is the energy harvesting efficiency, ${\mathbf{G}}_{\mathrm{CN}}=\mu P_t{\left|\gamma \right|}^2{\mathbf{a}}_r\left({\theta }_{ERN}\right){\mathbf{a}}_r^H\left({\theta }_{ERN}\right)$.

C4. SDR-BD algorithm

As we want to optimize the sensing performance of ISWSCPTS while meeting the basic requirements for communication and power transfer, the problem of beamforming design can be expressed as:

$$P_2:\begin{array}{cc}\hfill \underset{\eta }{max}& \hfill {\eta }^{\mathrm{H}}{\mathbf{G}}_{\mathrm{SN}}\eta \hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill {\eta }^{\mathrm{H}}{\mathbf{G}}_{\mathrm{CN}}{\eta }^{\ge }\rho \hfill \\ \hfill & \hfill {\eta }^{\mathrm{H}}{\mathbf{G}}_{\mathrm{ERN}}\eta \ge \varrho \hfill \\ \hfill & \hfill {\eta }^{\mathrm{H}}\eta \le 1\hfill \end{array}$$

(52)

where $\rho$ represents the minimum SNR required for communication, $\varrho$ denotes the minimum energy requirement for triggering the energy harvesting process in ERN.

Due to the max operation and quadratic constraints, problem $P_2$ is non-convex optimization. In this work, $P_2$ is solved through SDR technique [17]. According to ${\mathbf{x}}^{\mathrm{H}}\mathbf{A}\mathbf{x}=\mathrm{Tr}\left(\mathbf{A}\mathbf{x}{\mathbf{x}}^{\mathrm{H}}\right)$, where $\mathrm{Tr}(•)$ denotes the trace operation, $P_2$ can be transformed into the following equivalent optimization problem:

$$P_3:\begin{array}{cc}\hfill \underset{\Gamma }{max}& \hfill \mathrm{Tr}\left({\mathbf{G}}_{\mathrm{SN}}\Gamma \right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill \mathrm{Tr}\left({\mathbf{G}}_{\mathrm{CN}}\Gamma \right)\ge \rho \hfill \\ \hfill & \hfill \mathrm{Tr}\left({\mathbf{G}}_{\mathrm{ERN}}\Gamma \right)\ge \varrho \hfill \\ \hfill & \hfill \mathrm{Tr}\left(\Gamma \right)\le 1\hfill \\ \hfill & \hfill \mathrm{rank}\left(\Gamma \right)=1\hfill \\ \hfill & \hfill \Gamma ⪰\left(0\right)\hfill \end{array}$$

(53)

where $\Gamma =\eta {\eta }^{\mathrm{H}}$.

Then, the rank constraint is dropped to obtain the following relaxed optimization problem:

$$P_4:\begin{array}{cc}\hfill \underset{\Gamma }{max}& \hfill \mathrm{Tr}\left({\mathbf{G}}_{\mathrm{SN}}\Gamma \right)\hfill \\ \hfill \mathrm{s}.\mathrm{t}.& \hfill \mathrm{Tr}\left({\mathbf{G}}_{\mathrm{CN}}\Gamma \right)\ge \rho \hfill \\ \hfill & \hfill \mathrm{Tr}\left({\mathbf{G}}_{\mathrm{ERN}}\Gamma \right)\ge \varrho \hfill \\ \hfill & \hfill \mathrm{Tr}\left(\Gamma \right)\le 1\hfill \\ \hfill & \hfill \Gamma ⪰\left(0\right)\hfill \end{array}$$

(54)

The objective function and constraints in $P_4$ are all affine, thus making $P_4$ a convex optimization problem. $P_4$ can be solved by Matlab using the convex optimization toolbox CVX[]. If the rank of the solution ${\Gamma }^{★}$ obtained in $P_4$ is not equal to 1, then feasible solutions ${\eta }^{★}$ need to be extracted from ${\Gamma }^{★}$. Following [1], ${\eta }^{★}$ can be obtained as follows:

$${\eta }^{★}=\sqrt{{\lambda }_0}{\mathbf{q}}_0$$

(55)

where ${\lambda }_0$ represents the maximum eigenvalue of ${\Gamma }^{★}$, and ${\mathbf{q}}_0$ denotes the corresponding eigenvector.

3. Results and Discussions

This section conducts simulations to evaluate the performance of CP-OTFS MF-TDaPE , CP-OTFS AFS, and SDR-BD. The basic simulation parameters are summarized in Table 1.

3.1. Simulation of CP-OTFS MF-TDaPE

To begin with, the target detection is performed using CP-OTFS MF-TDaPE algorithm. The target range and velocity are 689.523 m and 95.054 m/s, corresponding to $l_p=46$ and $k_p=5$, respectively, with a TSC of $0.8247+0.4709i$. The SNR of this simulation is 10 dB. Figure 3 illustrates the output ${\xi }_{SN}$ of the MF, where a significant peak appears at positions $l=46$ and $k=5$. The target can be correctly detected using the CFAR algorithm.

To validate the parameter estimation performance of the CP-OTFS MF-TDaPE, we conduct parameter estimation for fast-moving targets with different velocities. The benchmark is OFDM waveform, and the estimation method is the fast Fourier transform (FFT) method described in [18]. Since both CP-OTFS and OFDM can accurately estimate the range, the main comparison focuses on velocity estimation performance and TSC estimation performance. The root mean square error (RMSE) is adopted as the performance metric, calculated as follows:

$$RMSE={\left(\frac{{\sum }_{n=1}^{Ns}{|\widehat{y}-y|}^2}{N_s}\right)}^{\frac{1}{2}}$$

(56)

where $\widehat{y}$ is the estimated parameter, y is the true value, $N_s$ is the number of samples.

Figure 4 and Figure 5 depict the RMSE results for velocity and TSC estimation of fast-moving targets with different velocities, respectively. It is evident that for fast-moving targets, the performance of velocity estimation sharply deteriorates with increasing velocity in OFDM, while OTFS consistently maintains accurate estimation. Regarding TSC estimation, although OFDM does not exhibit sensitivity to velocity variations, its estimation results still pale in comparison to the nearly error-free estimates provided by CP-OTFS. The results presented demonstrate the robustness of the ISWSCPTS framework in parameter estimation for high-speed motion scenarios.

3.2. Simulation of CP-OTFS AFS

The initial waveform symbols for this simulation are quadrature phase shift keying (QPSK) symbols. We first compare the AF of the waveform optimized by CP-OTFS AFS with the initial waveform. Figure 6 and Figure 7 respectively depict the AF of the initial and optimized waveforms. It can be observed that the optimized waveform exhibits a reduction of approximately 2-3 dB in side lobes in comparison to the initial waveform. Figure 8 illustrates the variation of the ISL with the number of iterations, showing a gradual decrease with successive iterations of the algorithm. However, the convergence speed of CP-OTFS AFS is slow, and there is still potential for improvement.

In fact, the performance of CP-OTFS MF-TDaPE in estimating TSC deteriorates with the emergence of interference. To illustrate this point, we introduced an interference target in the simulation scenario with delay and Doppler taps of 30 and 3, respectively, and the TSC of 1 for this target. Figure 9 shows the variation of the target TSC with SNR in the absence of interference, indicating similar performance between the initial QPSK waveform and the optimized waveform. Figure 10 illustrates the TSC estimation performance in the presence of interference. It is evident that the performance degradation appears. The reason is that the interference target affects TSC estimation through AF sidelobes. At the same time, it can also be observed that starting from an SNR greater than 6 dB, the optimized waveform exhibits stronger resistance to interference.

3.3. Simulation of SDR-BD

This simulation showcases the beam patterns of designed by SDR-BD under different system configurations and QoS settings. The specific simulation parameters for this study are detailed in Table 2.

Figure 11 illustrates the beamforming design results of SDR-BD under different configurations of $N_t$. The simulation result demonstrates that as $N_t$ increases, the beamforming becomes narrower in the directions of the target, CN, and ERN, indicating stronger directivity. Additionally, the beamforming exhibits lower sidelobes. Particularly, the highest sidelobe level of the beamforming near the target with $N_t=64$ is approximately 6.5 dB lower than that with $N_t=16$. This phenomenon arises from the increase in system antenna aperture with $N_t$, resulting in improved angular resolution of the system.

Figure 12 compares the beamforming with $N_t=16$ designed by SDR-BD under different communication and PT QoS settings. The simulation result indicates that increasing the QoS for communication and PT inevitably leads to a deterioration in sensing capability. Specifically, compared to the case where $\rho =15dB$ and $\varrho =5mW$, an increase of 5 dB in $\rho$ and 5 mW in $\varrho$ results in a reduction of approximately 4.8 dB in the antenna gain in the direction of the target. This outcome suggests that the beamforming designed by SDR-BD achieves a trade-off among sensing, communication, and PT in ISWSCPTS. Therefore, in practical application scenarios, the overall system performance optimization can be achieved according to respective QoS requirements.

4. Conclusions

In this work, the ISWSCPTS framework, which integrates ISACS, SWIPT, and OTFS, is proposed. ISWSCPTS comprises two work modes, SM and JWM, which utilize a single CP-OTFS waveform. SM is employed for target detection and parameter estimation, while JWM is used for target tracking, communication, and PT. For target detection and parameters estimation, CP-OTFS MF-DaPE algorithm is proposed, demonstrating superior performance in high-speed scenarios compared to OFDM. To enhance the robustness of parameters estimation against interference, CP-OTFS AFS algorithm is employed for waveform design. The waveform designed by CP-OTFS AFS reduces the ISL of corresponding AF and improves interference resilience. The SDR-BD algorithm is proposed to enhance the overall perceptual capability of the system by designing beamforming to maximize perceptual capability under communication and PT QoS constraints.

However, this work still has the following limitations: (1) The ISWSCPTS scenarios include only single target, CN, and ERN; (2) The CP-OTFS AFS incurs significant computational burden. In future work, we will focus on extending ISWSCPTS to multiple targets, CNs, and ERNs scenarios, and optimizing CP-OTFS AFS to improve its convergence speed.